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Quantum Painleve-Calogero Correspondence
A. Zabrodin
Moscow, September 20, 2011

(Based on joint work with A. Zotov)


Painleve equations



Compatibility of linear problems

Zero curvature condition

is equivalent to Painleve equation


Example (Painleve I)


Hamiltonian structure There is a time-dependent Hamiltonian H(p, q; T) such that Painleve equation is equivalent to

H(p, q; T) is a polynomial in p,q (K.Okamoto)


Painleve-Calogero correspondence
(P.Painleve, 1906; Yu. Manin, 1996; A.Levin and M.Olshanetsky, 1997; K.Takasaki, 2001) There exists a change of variables

(q,T)

(u,t)

such that the Painleve equation acquires the form of the Newton equation with Hamiltonian




Resembles a particular case of elliptic Calogero or Inozemtzev model


Quantum Painleve-Calogero correspondence emerges when we apply this change of variables to the auxiliary linear problems and supplement it by an appropriate change of the spectral parameter


Example (Painleve I)




Painleve equation is represented as compatibility condition for two scalar linear problems of the form

(A similar observation was made by B.Suleimanov, 1994)




The non-stationary Schrodinger (or Fokker-Planck) equation

is one of the equations of the Lax pair for Painleve, the one defining the time evolution, and, simultaneously, quantization of the Painleve equation! Having this equation, one does not need the second element of the Lax pair because it already reproduces Painleve by classical limit.


For Painleve VI with special choice of parameters we obtain the non-stationary Lame equation

which appears in conformal field theory and theory of XYZ spin chain.


Quantum Painleve-Calogero correspondence: conclusion

Painleve side

Calogero side
Quantum Calogero-like model in a non-stationary state

Auxiliary linear problem for Painleve

Quantization is equivalent to linearization (passing to auxiliary linear problem)